Since you are using a Turbo Filter instead of a regular filter, it is tied to the overall context, not to a specific appender. So, you'll need to define it outside: e.g. after the tag.
If you want to use it on a specific appender, you could use a regular filter.
<appender name="FILE" class="ch.qos.logback.core.rolling.RollingFileAppender">
<evaluator class="ch.qos.logback.classic.boolex.OnMarkerEvaluator">
<marker>PROCESS_INT</marker>
</evaluator>
... ... etc. ... ...
To check for that marker and allow or disallow for that appender.
For the formatting, when you log the message inside Java, instead of the "%s" style formatting, you need to use "{}".
LOGGER.debug(PROCESS_INTEGRATION_MARKER, "Method started:{}", "sendRequest");
Essentially, this is pretty similar to formatting the message yourself, but it could have performance benefits in some situations.